Theorem qsqeqor 10759

Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)

Assertion

Ref	Expression
qsqeqor	⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Proof of Theorem qsqeqor

Step	Hyp	Ref	Expression
1		qre 9716	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
2	1	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3		simplr 528	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4		qre 9716	. . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
5	4	ad3antlr 493	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7		sq11 10721	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
8	2, 3, 5, 6, 7	syl22anc 1250	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9		orc 713	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
10	8, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
11		oveq1 5932	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
12	11	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13		oveq1 5932	. . . . . . . . 9 ⊢ (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
14	13	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15		qcn 9725	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16		sqneg 10707	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
17	15, 16	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
18	17	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
19	14, 18	eqtrd 2229	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
20	19	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
21	12, 20	jaod 718	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
22	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
23	10, 22	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
24	17	eqeq2d 2208	. . . . . 6 ⊢ (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
25	24	ad3antlr 493	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
26	1	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27		simplr 528	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28		qnegcl 9727	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29		qre 9716	. . . . . . . . 9 ⊢ (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
30	28, 29	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
31	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
33	4	le0neg1d 8561	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
34	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
35	32, 34	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36		sq11 10721	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
37	26, 27, 31, 35, 36	syl22anc 1250	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38		olc 712	. . . . . 6 ⊢ (𝐴 = -𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
39	37, 38	biimtrdi 163	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
40	25, 39	sylbird 170	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
41	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
42	40, 41	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
43		0z 9354	. . . . . 6 ⊢ 0 ∈ ℤ
44		zq 9717	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
45	43, 44	ax-mp 5	. . . . 5 ⊢ 0 ∈ ℚ
46		qletric 10348	. . . . 5 ⊢ ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
47	45, 46	mpan 424	. . . 4 ⊢ (𝐵 ∈ ℚ → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
48	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
49	23, 42, 48	mpjaodan 799	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
50		qnegcl 9727	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51		qre 9716	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
52	50, 51	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
53	52	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54		simplr 528	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
55	1	le0neg1d 8561	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
56	55	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
57	54, 56	mpbid 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
58	4	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60		sq11 10721	. . . . . . . 8 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
61	53, 57, 58, 59, 60	syl22anc 1250	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
62	61	biimpd 144	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63		qcn 9725	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64		sqneg 10707	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
65	63, 64	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
67	66	eqeq1d 2205	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
68	67	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69		negcon1 8295	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
70	63, 15, 69	syl2an 289	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71		eqcom 2198	. . . . . . . 8 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵)
72	70, 71	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
73	72	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
74	62, 68, 73	3imtr3d 202	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
75	74, 38	syl6 33	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
76	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
77	75, 76	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
78	52	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79		simplr 528	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
80	55	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
81	79, 80	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
82	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
84	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
85	83, 84	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86		sq11 10721	. . . . . . 7 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
87	78, 81, 82, 85, 86	syl22anc 1250	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
88	65, 17	eqeqan12d 2212	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
89	88	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
90	63	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
91	15	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
92	90, 91	neg11ad 8350	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵))
93	87, 89, 92	3bitr3d 218	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
94	93, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
95	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
96	94, 95	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
97	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
98	77, 96, 97	mpjaodan 799	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
99		qletric 10348	. . . 4 ⊢ ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
100	45, 99	mpan 424	. . 3 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
101	100	adantr 276	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
102	49, 98, 101	mpjaodan 799	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2167   class class class wbr 4034  (class class class)co 5925  ℂcc 7894  ℝcr 7895  0cc0 7896   ≤ cle 8079  -cneg 8215  2c2 9058  ℤcz 9343  ℚcq 9710  ↑cexp 10647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-seqfrec 10557  df-exp 10648

This theorem is referenced by:  4sqlem10  12581